If d gcd a b and m lcm a b prove that dm abSolutionLet c

If d = gcd (a, b) and m = lcm (a, b), prove that dm = |ab|.

Solution

Let c = gcd(a,b)
Then a = c*x for some x, and b = c*y for some y.
x and y are coprime by defintion of gcd.

By definition, lcm(a,b) is divisible by a=c*x and b=c*y, therefore
lcm(a,b) = c*x*y (since x and y are coprime)

lcm(a,b)*gcd(a,b) = d * m = (c^2)*x*y = (c*x)*(c*y) = a*b